Suppose we wish to perform full Markowitz portfolio optimization for the S&P500, with or without private information
How many parameters would we need?
| Parameter | General Number | S&P Number |
|---|---|---|
| \(\sigma_{i}^{2}\) | Number of Stocks (\(N\)) | 500 |
| \(E(r_{i})\) | \(N\) | 500 |
| \(\sigma_{i,j}\) | \(\frac{1}{2} \times N \times (N-1)\) | 124,750 |
| Total | \(\frac{1}{2} \times N \times (N+3)\) | 125,750 |
We can use a CAPM/single-factor relation to simplify inputs
We begin with the CAPM regression (note: BKM call this the “Single Index Model”)
\[ r_{i,t} - r_{f} = \alpha_{i} + \beta_{i}(r_{m,t} - r_{f}) + \epsilon_{i,t} \]
\[ \epsilon_{i,t} \sim \mathcal{N}(0, \sigma_{i}^{2}(\epsilon)) \]
Degree of idiosyncratic variance \(\sigma_{i}^{2}(\epsilon)\) is allowed to vary for different stocks
Additionally, we assume that \(Cov(\epsilon_{i,t}, \epsilon_{j,t}) = cov(\epsilon_{i,t}, r_{m,t}) = 0\)
What does this buy us?
\[ E(r_{i} - r_{f}) = \alpha_{i} + \beta_{i}E(r_{m,t} - r_{f}) \]
\[ \sigma_{i,j} = cov(r_{i}, r_{j}) = \beta_{i}\beta_{j}\sigma_{m}^{2}, \; i \not= j \]
\[ \sigma_{i}^{2} = \beta_{i}^{2}\sigma_{m}^{2} + \sigma_{i}^{2}(\epsilon) \]
| Parameter | General Number | S&P Number |
|---|---|---|
| \(E(r_{m})\) | 1 | 1 |
| \(r_{f}\) | 1 | 1 |
| \(\sigma_{m}^{2}\) | 1 | 1 |
| \(\alpha_{i}\) | \(N\) | 500 |
| \(\beta_{i}\) | \(N\) | 500 |
| \(\sigma_{i}^{2}\) | \(N\) | 500 |
| Total | \((N+3)\) | 1,503 |
A lot easier than 125K parameters!
Pop Quiz: If alphas are all zero, and all assets are included, what portoflio do we choose?
Single index mocel brings both parsimony and discipline to inputs
CAPM presents an equilibrium view of expected returns vs. reliance on messy historical returns
What do we miss by doing this?
This is almost the Treynor-Black Model
So far, we have looked at a small number of assets for convenience
In reality, however, even the most diligent analysts will only have valuable views on a handful of assets
How do we achieve benefits of diversification while still focusing our views on a handful of stocks?
\[ w_{i}^{0} = \frac{\alpha_{i} / \sigma^{2}_{i}(\epsilon)}{\sum_{j} \alpha_{j} / \sigma^{2}_{j}(\epsilon)} \]
\(w_{i}^{0}\) is the optimal weight to put on stock \(i\) within the active portfolio \(A\)
\(\alpha_{i}\) is the expected excess return over the CAPM prediction for stock \(i\)
\(\sigma_{i}^{2}(\epsilon)\) is the residual (idiosyncratic) variation of stock \(i\)
So, how do we interpret this equation?
Next decide how much to put in the active portfolio, \(w_{A}\)?
Solving this analytically yields
\[ w_{A} = \frac{\alpha_{A} / \sigma_{A}^{2}(\epsilon)}{E(r_{m}-r_{f})/\sigma_{m}^{2} + (1-\beta_{A})(\alpha_{A}/\sigma_{A}^2(\epsilon))} \]
where \[ \beta_{A} = \sum_{i \in A} w_{i}\beta_{i}, \qquad \alpha_{A} = \sum_{i \in A} w_{i}\alpha_{i}, \qquad \sigma^{2}_{A}(\epsilon) = \sum_{i \in A} w_{i}^{2}\sigma^{2}_{i}(\epsilon) \]
Final active portfolio weight on stock \(i\) is \(w_{i}^{A} \times w_{A}\)
\((1- w_{A})\) is invested in the passive index
Market returns are 9% with a variance, \(\sigma_{m}^{2}\) of 4% and a risk-free rate of 3%
Combine the index with the following beliefs regarding Facebook and Twitter:
| \(\alpha\) | \(\beta\) | \(\sigma_{i}^{2}(\epsilon)\) | |
|---|---|---|---|
| FB | 0.5% | 2 | 7% |
| TWTR | 0.2% | 1.75 | 12% |
What are we assuming about the other stocks?
Plugging in to get active weights yields:
\[ w_{FB}^{0} = \frac{ \alpha_{FB} \big/ \sigma_{FB}^{2}(\epsilon)}{ \alpha_{FB} \big/ \sigma_{FB}^{2}(\epsilon) + \alpha_{TWTR} \big/ \sigma_{TWTR}^{2}(\epsilon)} = 0.81, \qquad w_{TWTR}^{0} = 0.19 \]
\[ \begin{aligned} \beta_{A} &= 0.81 \times 2 + 0.19 \times 1.75 = 1.95\\ \alpha_{A} &= 0.81 \times 0.005 + 0.19 \times 0.002 = 0.0044\\ \sigma_{A}^{2}(\epsilon) &= 0.81^{2} \times 0.07 + 0.19^{2} \times 0.12 = 0.05 \end{aligned} \]
\[ w_{A} = \frac{\alpha_{A} / \sigma_{A}^{2}(\epsilon)}{E(r_{m}-r_{f})/\sigma_{m}^{2} + (1-\beta_{A})(\alpha_{A}/\sigma_{A}^2(\epsilon))} = \frac{0.0044/ 0.05}{0.06/0.04 + (1-1.95)(0.0044/ 0.05)} = 0.06 \]
So we invest roughly 6% in an active portfolio, and 94% in the index
Given the active portfolio weights in FB and TWTR, we get final portfolio weights of 94% for the index, 5% for FB, and 1% TWTR
Evidence for persistent superior performance is weak
Malkiel (1995) tracks funds based on above/below median performance
Alternatively, the active share (Cremers and Petajisto, 2009)
Decompose portfolio into two parts:
\[ \text{Portfolio} = \text{Index} + (\text{Portfolio} - \text{Index}) \]
\[ \text{Active Share}_{fund} = \frac{1}{2}\sum \left| w_{fund, stock} - w_{index, stock} \right| \]
Indicates the size of the active positions as a fraction of the entire portfolio
The Black-Litterman approach is much less fluffy.
Straightforward approach to incorporate three things:
Approach is somewhat technical, so we will first walk through the cookbook
Step 1:
Step 2:
Come up with an independent estimate of \(E(r)\), your “view”
Assign a level of uncertainty to this estimate (Omega \(\omega\))
Step 3:
Use Bayes’ rule to formulate posterior distribution of \(E(r)\) as a weighted average of views + equilibrium estimates, weighted by precision (inverse uncertainty)
Proceed with MVE based on refined view of expected returns and covariance matrix